We're interested in the probability that voting B over C will change the winner from C to B. That's what I mean by Pbc, for the purposes of this problem. Examining the ways that voting for A can change Pbc shows that voting for A can only make it less likely that voting B over C will change the winner from C to B: Say B & C are tied before we vote for B, making B win. Say our vote for A promotes A to the vote total of the tied B & C. Without that A vote, and without our vote for B, it's 50% that C wins. And so it's 50% that our vote for B over A changes the winner from C to B. But say we vote for A & thereby promote A to the same vote total as that initial BC tie. Now, without our vote for B over C, it's 1/3 that C wins. And so it's only 1/3 that our vote for B over C changes the winner from C to B. Say that, instead, our vote for A promotes A from the level at which B & C are tied before I vote for B over C, to the level 1 vote above that. In that case, the winner before I casts my B vote is A, and there's zero probability that doing so changes the winner from C to B. It might change it from A to B, but not from C to B. Or maybe A & B are one vote below C. My vote for B changes that to a BC tie. I change the winner from C to B with a probability of 1/2. Say, instead that I've voted for A too, so now I'm changing an AC tie into an ABC tie. There's 1/2 probability that C was the winner before I voted B over C, and there's only 1/3 probability that B is the winner after I vote B over C. This can be summarized by pointing out that wherever I make A's vote total arrive at, to make some tie, it has one effect: It reduces the likelihood that some particular other member of that tie wins. Changing the winner from C to B requires that first C win and then B win. Making A arrive where one of those is can only have the effect of making it less likely for one of those to win, either before or after I vote B over C, or both. And of course if my promotion of A makes it the initial winner, then too I'm not changing the winner from C to anything, because C isn't the winner before I vote for B. A is the winner before I vote for B. So the only thing that A can do when it arrives is reduce the probability of B or C winning, and that reduces Pbc overall. Mike Ossipoff _________________________________________________________________ Get your FREE download of MSN Explorer at http://explorer.msn.com
